310 XV NORMED ALGEBRAS AND SPECTRAL THEORY j£ such that (fn) is a Cauchy sequence in A, show that the sequence (o)/n) converges in &l, and that the sequence of classes of the functions \fn\2<**fnl converges in L1.) (f) Let In be the interval [— «, n] in R and stf \ !„ the set of functions belonging to <s# whose support is contained in IB . Show that, for each/e j/, the sequence of functions fipin tends to /in J#. The topologies defined on d \ In by NB and by the seminorm N2 are equivalent. Deduce that the space Jf(R) is dense in ^/, and that the Banach space A is separable. (g) If /and g are two functions belonging to «^, show that g ~ OJf O)g Deduce that NB(/*#) ^NB(/)NB(#) and hence that A is a commutative Banach algebra (called Beurling's algebra}. It has no unit element. 5. Let A be a normed algebra with no unit element. On the set A ~ A x C, define a C-algebra structure as follows: (x, AX*', A') = (xxf 4- A*' 4- A'x, AA'), H(x, A) *= QJLX, /nA). Show that (0, 1) is the unit element of A, and that \\(x, A)|| == \\x\\ 4- |A| is a norm on A for which A is a normed algebra, and a Banach algebra if A is a Banach algebra. 2. SPECTRUM OF AN ELEMENT OF A NORMED ALGEBRA In this section we shall assume that A is a normed algebra with unit element e ^ 0. For each x e A, a complex number C is said to be a regular value for x if x — £e has an inverse in A. The complex numbers C that are not regular values for x are called the spectral values of x. The set of spectral values of x is called the spectrum of x (in A) and is denoted by SpA(x) or Sp(^:). When A is the algebra jSf (E) of continuous endomorphisms of a normed space E ^ {0}, these definitions agree with those of (11,1). Examples (15.2.1) For each A e C, it follows from the definition that SpA(^e) = {A} and that SpA(jc + Ae) = SpA(x) +. A, for all x e A. (15.2.2) Every C e C such that x — £e is a (left or right) zero-divisor in A is a spectral value of x, but the converse is not true (Section 11.1, Problem 3).sing to the